Concrete/Steel Bond in Reinforced Concrete Structures Subjected to Dynamic Loadings: Basis of New Numerical Model

Abstract

This paper presents a new approach for modeling macrocrack propagation in reinforced concrete structures under both static and dynamic loading conditions. The numerical modeling is based on (1) the use of a probabilistic semi-explicit cracking (PSEC) model for macrocrack propagation and (2) the use of a deterministic damage model for the bond between steel and concrete. Another distinctive feature of the proposed modeling approach is the exclusive use of linear volumetric finite elements, both for macrocrack propagation and for the concrete/steel bond. For the latter, a single layer of volume elements is used along the reinforcement bars. Furthermore, the paper details a methodology for incorporating strain rate effects into the bond model under dynamic loading. It also outlines procedures for identifying the parameters required for both the static and dynamic formulations of the proposed models.

1. Introduction

Modeling the interaction between concrete and steel, while incorporating the non-linear behavior associated with cracking near the reinforcement, is essential for evaluating crack widths and spacing in reinforced concrete elements. Over time, numerous research efforts and modeling approaches have been proposed to tackle this issue [1,2,3,4,5,6,7,8,9,10,11,12]. These bond models vary depending on the cracking model used for the concrete.

What sets the concrete/steel bond model presented in this study apart is its foundation on a probabilistic approach to concrete cracking. Historically, two primary types of probabilistic cracking models have been introduced:

(1) The probabilistic explicit cracking (PEC) model, which has been developed and validated for a long time [13]. Its main drawback is that it uses interface elements to represent cracks, requiring a dense nodal network, as these non-linear elements must connect all volume elements in the mesh. Consequently, this leads to high computational costs, particularly when simulating large-scale structures. (2) To overcome this limitation, a probabilistic semi-explicit cracking (PSEC) model was later developed. This method uses linear volume elements to simulate the development of macrocracks. To ensure consistency in the modeling scale, the bond between concrete and steel was also modeled using a single layer of volume elements. This choice of modeling the concrete/steel bond is original compared to most other approaches in the literature, which tend to favor the use of special interface elements. A preliminary version of such a bond model based on volume elements was introduced in an earlier work [14].

The core innovation of the present study lies in the development of this volume element-based concrete/steel bond model tailored for dynamic analysis.

2. PSEC Model and Concrete/Steel Bond Model in Statics

The PSEC model has been extensively detailed in prior research [15,16]; a brief summary is provided in this paper. The core principles of the PSEC model can be outlined as follows:

• Each linear volume element is assumed to represent a heterogeneous material volume. The level of heterogeneity is defined as the ratio between the element’s volume and that of the largest aggregate in the concrete.
• The mechanical behavior of each element is volume-dependent, with elementary mechanical properties randomly assigned across the computational mesh. These properties follow spatially uncorrelated random fields, causing the statistical characteristics to vary from one element to another, depending on the heterogeneity ratio.
• Macrocrack propagation is modeled as the result of the coalescence of randomly generated elementary voids—each corresponding to the disappearance of a volume element.

These characteristics justify the term “semi-explicit” used to describe the model.

At the finite element scale, the energy effects related to elementary cracking are captured using a simple isotropic damage law governed by a single scalar parameter [15,17,18,19,20,21]. For simplicity, the stress–strain relationship is expressed using a bilinear formulation. The first linear segment represents the material’s elastic behavior up to the peak, while the second linear segment models its non-linear (post-peak) response. Additionally, a probabilistic energy-based regularization is incorporated

Without delving into the numerical implementation details of the model (which uses an implicit algorithm), its main features can be summarized as follows:

• Elementary cracking is represented through the stress–strain relationship. The dissipative process (i.e., crack propagation within a finite element) begins when the major principal stress at a specific Gauss point (located at the center of the element) reaches the material’s tensile strength, f_t. Dissipation progresses according to the positive component of the strain projected in the direction of the major principal stress, n_σ. Once the total energy available in the element is fully dissipated (i.e., when the damage variable D reaches 1), the element is considered cracked, and its stiffness matrix is set to zero, effectively creating a void. This prevents stress-locking effects.
• The model is implemented numerically using a rotating crack approach [22,23]. During the dissipative phase (D < 1), the direction nσ may evolve in response to changes in the stress state. Once the element is fully cracked (D = 1), the direction of the crack plane becomes fixed: n^c = n_σ.
• A crack (modeled as a void) is only considered to exist once D = 1. Before this threshold is reached, the energy dissipation within the element is not directly linked to any physical microcracking. Therefore, crack opening can only be assessed once the damage variable reaches 1. The crack opening is determined by projecting the element’s displacement in the direction of n^c.
• Crack reclosure is not explicitly handled. The model assumes that the dissipative process does not affect the element’s stiffness under compression. Consequently, if a crack recloses, the stiffness in compression is fully restored, while the tensile strength f_t is reduced to zero.
• Linear volume elements are used in the numerical model. This choice is motivated by the goal of applying the semi-explicit approach to large-scale structures, where reducing computation time is crucial. Linear elements are preferred over more complex quadratic ones for their simplicity and efficiency.

In summary, the constitutive law of the model is governed by two key parameters: the tensile strength (f_t) and the volumetric energy dissipation density (g_c). An energetic regularization technique enables the computation of g_c from the surface cracking energy G_c using the relation: g_c = G_c/l_e. In this context, G_c refers to the critical strain energy release rate as defined by Irwin [24] in linear fracture mechanics (LFM), and l_e denotes the characteristic length of the element, calculated as l_e = V ^(1/3), where V is the volume of the finite element.

The mechanical properties f_t and G_c are assigned to each element based on spatially uncorrelated random distributions: a Weibull distribution for f_t and a lognormal distribution for G_c. As previously discussed, the statistical parameters of these distributions depend on the volume of each finite element through the local heterogeneity ratio. The sole exception is the mean value of G_c, which is considered independent of the element’s volume. As a result:

• The energy required to initiate and propagate a macrocrack can, on average, be regarded as a material-specific parameter, determined only by the concrete type.
• Because of the material’s inherent heterogeneity—more prominent at the finite element scale—the dissipated energy displays a degree of randomness. This variability is directly related to the size of the loaded volume, which may or may not include obstacles to crack propagation.

The PSE model is illustrated in Figure 1.

Regarding the determination of the mechanical properties of the material, several experimental and theoretical studies [16] have proposed and validated relationships that enable the estimation of both the mean value and standard deviation related to f_t and G_c for all types of concrete. These values depend solely on two parameters: the concrete’s compressive strength and the maximum aggregate size.

Within the framework of this model, the interface zone between concrete and reinforcement is represented by a single layer of volume elements connecting the concrete to the rebar. This layer serves the following purposes:

• To simulate local failure at the steel–concrete interface when shear-induced cracking leads to a loss of adhesion.
• To represent sliding between the concrete and the rebar prior to complete interface failure.
• To account for local friction between concrete and steel following interface failure.

To fulfill these roles within the context of the probabilistic semi-explicit cracking model described earlier, the following approach is adopted for modeling the bond between the rebar and the concrete:

• A single layer of volume elements is used to represent the bond zone between the two materials.
• Two types of mechanical behavior are considered for these elements: one corresponding to macrocrack propagation (Mode I) through the rebar, and the other to the bond behavior (shear response) at the interface.
• Each volume element is assigned to only one type of behavior. That is, a given element either simulates the propagation of a macrocrack through the rebar or models the bonding behavior between concrete and steel.

If a volume element is associated with macrocrack propagation, it follows the same modeling approach as defined in the PSEC model.

If, instead, the volume element is assigned to represent bond behavior, the concrete/steel bond is modeled as an interface zone that gradually deteriorates under shear loading (tensile failure is not considered). During this degradation phase, and before complete failure of the interface, stress transmission between the steel and concrete is still assumed to occur.

It is important to note that both the tensile and shear damage criteria are evaluated at the centroid of each volume element.

A simplified approach is adopted, based on a damage model that maintains a constant stress level once the critical shear threshold is reached (see Figure 2). When the relative tangential strain—measured in the direction parallel to the rebar surface—exceeds a predefined critical value, the volume element is considered to have failed.

The damage model used to represent the concrete/steel bond is considered deterministic. This assumption is justified by observations showing that the cracking process around the rebar is mainly governed by the mechanical interaction with the rebar ribs, rather than by the heterogeneity of the surrounding concrete.

The evolution of damage is described as follows:

$$\left\{\begin{array}{c} d=0 \left|{\epsilon }_{tg}\right|<{\epsilon }_{tg}^e \\ d={\displaystyle \frac{1-{\epsilon }_{tg}^e}{\left|{\epsilon }_{tg}\right|}} if {\epsilon }_t^e<\left|{\epsilon }_{tg}\right|<{\epsilon }_{tg}^c \\ d=1 \left|{\epsilon }_{tg}\right| \ge {\epsilon }_{tg}^c \end{array}\right.$$

(1)

where ε_tg^e represents the threshold of the tangential elastic strain in the direction parallel to the surface of the rebar, ε_tg^c denotes the critical tangential strain, and |ε_tg| is the parameter that governs the damage evolution of the cohesion stress C (critical tangential stress). To ensure the positivity of thermodynamic dissipation, the damage can only increase. This can be summarized as follows:

$$\left\{\begin{array}{c}\dot{d} \ge 0 \\ d=\max \left(d_{0 }, d\right)\end{array}\right.$$

(2)

where d₀ is the initial damage state and d is the current damage state.

In this study, only the values of the maximum shear stress (C) and the tangential critical strain (ε_tg^c) need to be determined.

As outlined in previous sections, each volume element within the bond zone is governed by a single behavioral law. In other words, an element is designated either for macrocrack propagation or for modeling sliding with friction along the rebar. To enforce this, the following procedure is applied at each calculation increment to evaluate whether the element is more critically affected by tensile loading (Mode I) or by shear. Two situations are possible:

• If only one of the two failure criteria is satisfied within the element, it will be assigned accordingly—as either a tension or shear element—for the remainder of the simulation.
• If both criteria are satisfied, the element will follow the mode (tension or shear) associated with the greater margin between the applied stress (tensile or shear) and its corresponding strength (f_t or C), and this behavior will remain fixed throughout the simulation.

3. Determination of the Mechanical Parameter Values Related to the Shear Behavior of the Bond Elements

Previous research [25] that employed interface elements to model the bond between concrete and rebar proposed a method for determining the mechanical parameters of these interface elements by calibrating them against experimental results from tie-beam tests. This approach has been validated in subsequent studies [25].

In the present work, we adopt the same strategy to characterize bond behavior, focusing specifically on the mechanical response of a tie-beam.

There are two possible scenarios for implementing this inverse identification method:

• Experimental data available: Experimental results from tie-beam tests involving the same concrete and reinforcement in the structure being analyzed are available. This is the ideal case as it allows for direct calibration.
• No experimental data available: In the absence of physical test results, the inverse method can still be performed using numerical tie-beam simulations. In this case, validated local modeling is employed to replicate the test virtually, using the following approach:

  ○

  The rebar is modeled with its actual ribbed geometry.

  ○

  The cracking process in the surrounding concrete is simulated using the probabilistic semi-explicit cracking model (see Section 2). A highly refined mesh is applied around the rebar—at the scale of the ribs—to capture the detailed cracking, sliding, and friction behavior. This numerical technique has already proven successful in prior studies [25].

Key considerations for the inverse approach include:

• The macroscopic behavior of the tie-beam—particularly the load–displacement curve (i.e., the relative displacement between the rebar and concrete at one end)—is not sufficient by itself to identify a unique set of values for the maximum shear stress (C) and the tangential critical strain (ε_tg^c). Additional criteria, such as the number of macrocracks observed in the numerical simulation, must also be considered to refine the calibration.
• The identified values of C and ε_tg^c are dependent on the size of the volume elements. Therefore, it is essential to use the same element size in the simulation of the reinforced concrete structure as was used in the inverse identification process.

4. PSEC Model and Concrete/Steel Bond Model in Dynamics

PSEC model has been recently developed and validated for analyzing the cracking behavior of concrete structures subjected to impact loading. This development is detailed in [26]. The model accounts for strain rate effects on the material’s tensile response, incorporating micro-inertia and moisture influences via the Stefan effect [27]. Structural inertia is naturally captured within classical dynamic analysis frameworks [28,29,30,31,32,33,34] through the inclusion of mass and damping matrices in the equation of motion. A standard dynamic analysis is conducted using the Newmark-beta algorithm for time-step integration.

To incorporate strain rate effects within the PSEC model, several experimental studies focused on the Stefan effect have been considered [27]. These studies led to the following expression for dynamic tensile strength [26]:

$${\mathrm{f}}_{\mathrm{t},\mathrm{dyn}} = {\mathrm{f}}_{\mathrm{t},\mathrm{stat}} + 2.8 + 0.3 \ln \left(\dot{\sigma }\right)$$

(3)

where the tensile strength is given in MPa, and the stress rate has units of GPa/s. f_t,stat and f_t,dyn are, respectively, the static tensile strength and the dynamics tensile strength.

For the rate effect on fracture energy, G_IC, the following relationship has been proposed [26]:

G_ICdyn = G_ICstat (f_t,dyn/f_t,stat)

(4)

It is important to highlight that Equation (4) is based on sound physical reasoning. The cracking process in a uniaxial tension test specimen (governing f_t) is comparable to the process occurring at the tip of a propagating macrocrack (governing G_IC). As such, the strain rate effect is assumed to apply similarly in both cases.

With respect to the bond zone between steel reinforcement and concrete, as previously discussed, its non-linear behavior results from the initiation and evolution of microcracks prior to complete bond failure. This process is similar in nature to macrocrack propagation and the associated fracture process zone. Consequently, applying a similar rate-dependent formulation to the bond behavior is considered justified.

The following rate-dependent expressions are proposed for the bond parameters:

C_dyn = C_stat (f_t,dyn/f_t,stat)

(5)

And

ε_tg^c _dyn = ε_tg^c _stat (f_t,dyn/f_t,stat)

(6)

The treatment of the two criteria governing the steel–concrete interface behavior—tensile and shear—is managed in the same manner for dynamic loading as for static loading.

5. Conclusions and Perspective

This study focuses on the concrete/rebar bond behavior in a dynamic context, using a macroscopic approach designed to analyze the cracking processes in large-scale reinforced concrete structures. The methodology accounts for both macrocrack propagation through the reinforcement bars and sliding along the bars, which results from microcracking in the surrounding concrete.

To simulate macrocrack propagation, a probabilistic semi-explicit cracking (PSEC) model is introduced. This model, validated for both static and dynamic conditions, combines damage mechanics and linear fracture mechanics. It employs an implicit algorithm for static analysis and the Newmark-beta time integration algorithm for dynamic simulations.

To capture the non-linear bond behavior along the rebar, a simple deterministic damage model is proposed. This model is governed by two key parameters: the cohesion stress (C) and the critical tangential strain (ε_tg^c). The bond zone is modeled using a single layer of linear volume elements.

The dynamic analysis of the interface model represents the novel contribution of this work. In addition, the paper outlines the methodologies for determining all parameter values used in both static and dynamic formulations.

Looking forward, the numerical implementation of the interface model in a dynamic setting remains a critical step. This work provides the scientific and engineering communities with a solid foundation to support such developments. Following this implementation, a validation phase—based on available experimental data from the literature—will be necessary to confirm the accuracy and robustness of the proposed models.